The Model

The setting is an infinite-horizon, discrete-time environment in which a manager acts on behalf of shareholders to maximize the expected present value of their distributions. The firm uses capital in a decreasing-returns technology to generate operating income according to \(zK^{\alpha}\), where \(K\) is the capital stock, \(z\) is a profitability shock, and \(\alpha < 1\) governs the degree of returns to scale.

The profitability shock \(z\) is log-normally distributed and follows the process: \[ \ln(z') = \rho \ln(z) + \sigma \varepsilon', \quad \varepsilon' \sim \mathcal{N}(0, 1) \]

Each period, the firm chooses investment \(I\), which is defined by a standard capital accumulation identity: \[ K' = (1 - \delta)K + I \]

The price of capital goods is normalized to one. The firm’s cash flow, \(E^*(K, K', z)\), is its operating income minus its expenditure on investment:

\[ \underbrace{E^*(K, K', z)}_{\text{Internal cash flow}} = \underbrace{zK^{\alpha}}_{\text{Income}} - \underbrace{(K' - (1 - \delta)K)}_{\text{Investment}} \]

Cash flows to shareholders, \(E(K, K', z)\), are defined in terms of the firm’s (internal) cash flows \(E^*(K, K', z)\). A positive firm cash flow is distributed to its stockholders, while a negative cash flow implies that the firm instead obtains funds from shareholders. In this case, the firm pays a linear cost \(\lambda\). Thus:

The \(E\) function can also be thought of as the firm’s per-period return function.

\[ \begin{align*} E(K, K', z) = \begin{cases} E^* & \text{if } E^* \geq 0 \\ E^*(1 + \lambda) & \text{if } E^* < 0 \end{cases} \end{align*} \]

Having defined cash flows, we can now state the firm’s dynamic programming problem: \[ \begin{aligned} \Pi(K, z) = & \max_{K'} \left\{ E(K, K', z) + \beta \; \mathbb{E}[\Pi(K', z')] \right\} \\ &\text{s.t.} \quad K' = (1 - \delta)K + I \end{aligned} \]

Notice that this optimization problem does not have a closed-form solution, because of the non-linearity introduced by the cash flow function \(E(K, K', z)\). Therefore, we will solve the problem numerically using value function iteration.